Nesting is a useful technique in many areas of database practice. For instance, nesting is a fundamental operation for the nested relational data model, it can be applied to reduce the level of data redundancy in a database instance, to improve query processing or to convert data from one model to another. We further address the question when nesting operations commute with one another, i.e., when the final nested database relation is independent of the order in which the nesting operations are applied. In fact, it has been shown that the satisfaction of weakmultivalued dependencies provides a sufficient and necessary condition for the commutativity of nesting operations. We study inference systems for different notions of implication for weak multivalued dependencies. First, we establish an axiomatisation with the property that every weak multivalued dependency can be inferred either without any application of the complementation rule or by a single application of the complementation rule necessary only in the very last step of the inference. Consequently, the complementation rule is a mere means to achieve a decomposition of the database. Secondly, we drop the assumption of having a fixed underlying schema, and establish an axiomatisation of weak multivalued dependencies for the notion of implication in this context.
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